Find the Derivative

y=(t^2+3)e^t

use the product rule

dy/dt = (t^2 + 3)(e^t) + (e^t)(2t)
= e^t( t^2 + 2t + 3)

To find the derivative of the function y = (t^2+3)e^t, we can use the product rule. The product rule states that if we have a function f(x) multiplied by another function g(x), then the derivative of the product fg is given by (f'g + fg'), where f' is the derivative of f and g' is the derivative of g.

In this case, f(t) = t^2+3 and g(t) = e^t. Let's find the derivatives of f(t) and g(t) separately:

- Derivative of f(t):
To find the derivative of t^2+3, we differentiate each term separately. The derivative of t^2 is 2t (using the power rule), and the derivative of 3 is zero (a constant has a derivative of zero). Therefore, f'(t) = 2t.

- Derivative of g(t):
The derivative of e^t is simply e^t. The derivative of e^t is the same as the original function itself.

Now, applying the product rule, we have:

y' = (f'g + fg')
= (2t * e^t + (t^2+3) * e^t)
= (2te^t + (t^2+3)e^t)
= (2t + t^2 + 3)e^t

Therefore, the derivative of y = (t^2+3)e^t is y' = (2t + t^2 + 3)e^t.